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Wave-Equation Migration in Wave-Equation Migration in Anisotropic MediaAnisotropic Media
Jianhua YuJianhua Yu
University of UtahUniversity of Utah
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Contents Contents
Motivation
Anisotropic Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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Contents Contents
Motivation
Anisotropic Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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What Blurs Seismic Images? What Blurs Seismic Images?
Irregular acquisition geometry
Bandwidth source wavelet
Velocity errors
Higher order phenomenon: Anisotropy
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Anisotropic ImagingAnisotropic Imaging
Ray-based anisotropic migration: Anisotropic velocity model
Anisotropic wave-equation migration:
---Ristow et al, 1998
---Han et al. 2003
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Objective: Objective:
High efficiency
Improve image accuracy
Develop 3-D anisotropic wave-equation migration method in orthorhombic model
>78 wave propagatoro
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Contents Contents
Motivation
Anisotropy Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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General Wave EquationGeneral Wave EquationWave equation in displacement
il
kijkl
j
i Fx
uC
xt
u
)(2
2
Ui : displacement component
Cijkl : 4th-order stiffness tensor
3 3
3,2,1,,k l
jiklijklij ec
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Eigensystem EquationEigensystem Equation
0
3
2
1
2333231
232
2221
13122
11
U
U
U
V
V
V
Polarization components of P-P, SV, and SH waves
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Orthorhombic AnisotropicOrthorhombic Anisotropic2355
2266
211111 ncncncΓ
21661112 n)nc(cΓ
31551313 n)nc(cΓ
2344
2222
216622 ncncncΓ
2333
2244
215533 ncncncΓ
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Orthorhombic AnisotropicOrthorhombic Anisotropic
21662121 n)nc(cΓ
32442323 n)nc(cΓ
31553131 n)nc(cΓ
32443232 n)nc(cΓ
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Decoupled P plane Wave Motion Decoupled P plane Wave Motion Equations in (x,z) and (y,z) planesEquations in (x,z) and (y,z) planes
0)(
)(
2
1
2233
2555513
551322
552
11
U
U
KcKcKKcc
KKccKcKc
zxzx
zxzx
0)(
)(
3
2
2233
2444423
442322
442
22
U
U
KcKcKKcc
KKccKcKc
zxzx
zxzx
and
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Decoupled P plane Wave Motion Decoupled P plane Wave Motion Equations in (x,z) and (y,z) planesEquations in (x,z) and (y,z) planes
0)(
)(
2
1
2233
2555513
551322
552
11
U
U
KcKcKKcc
KKccKcKc
zxzx
zxzx
0)(
)(
3
2
2233
2444423
442322
442
22
U
U
KcKcKKcc
KKccKcKc
zxzx
zxzx
and
det
det
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Dispersion EquationsDispersion Equations
24)1()1(22
2)1(2242
)(2
)1(
x
xz K
KK
(x,z) plane
24)2()2(22
2)2(2242
)(2
)1(
y
yz K
KK
(y,z) plane
Thomsen’s Parameters
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33c
33
3322)1(
2c
cc
)(2
)()(
443333
24433
24423)1(
ccc
cccc
Thomsen’s Parameters
33
3311)2(
2c
cc
)(2
)()(
553333
25533
25513)2(
ccc
cccc
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VTI:
)2()1(
)2()1(
5544 cc
20
20
2
0 )](1[
)(
x
xm
A
KBB
KA
24000
20
2
200
242
)(2
)21(
x
xz K
KK
)11
(0
FFD algorithm
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FFD Anisotropy Migration
)21( )1(00 aA
)21( )1(aA
)1(0
)1(00 )(2)2(2 bababB
)1()1( )(2)2(2 bababB
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How to Set Velocity and Anisotropy Parameters
a & b : Optimization coefficients of Pade approximation for FD
d 0Velocity:
Anisotropy:
)1()1(0
)1( d
)1()1(0
)1( d
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0
5
Err
or %
0 90
Pade Approximation Comparison
Angle
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0
0.05
Error %
0 78
Pade Approximation Comparison
Angle
Beyond 78 within 0.02 %
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Contents Contents
Motivation
Anisotropy Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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0.6
0
Kz
Kx -0.3 0.3 Kx -0.3 0.3
Weak Anisotropy Strong Anisotropy
Exact Exact
** Approximation ** Approximation
2.01.0 00 4.05.0
00 00 0015.005.0
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0.3
0
Kz
Kx -0.3 0.3
Dispersion Equation Approximation
Strong anisotropy
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0
2.0
Dep
th (
km
)
V/V0=3
V/V0=3
iso
iso
New
Sta
nd
ard
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0
2.0
Dep
th (
km
)
V/V0=3
V/V0=3
Weak Aniso
Strong Aniso
2.01.0 00 4.05.0
00 00 0015.005.0
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Contents Contents
Motivation
Anisotropy Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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00
1
Dep
th (
km
)1.5X (km)
Velocity (2.0-3.0 km/s)Velocity (2.0-3.0 km/s)
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00
1
Tim
e (s
)
1.5X (km)
Velocity (2.0-3.0 km/s)
0 1.5
Anisotropic data (SUSYNLVFTI)
0
1.2
Tim
e (s
)
X (km)
Isotropic data (SUSYNLY)
1.004.0 00 00
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00
1
Dep
th (
km
)
1.5
X (km)
Isotropic data Isotropic mig (su)
0 1.5
Anisotropic data Isotropic mig
0 1.5
Anisotropic data Anisotropic mig
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Contents Contents
Motivation
Anisotropy Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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00
4
Dep
th (
km
)5X (km)
Salt Model (VTI)
1.0045.0 00 00
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00
4
Dep
th (
km
)5X (km)
Iso-mig
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00
4
Dep
th (
km
)5X (km)
VTI Aniso-mig
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0 1.5
Anisotropy Error 40 %
X (km)
0
4
Dep
th (
km
)
0 1.5
Anisotropy Error 10 %
X (km)0 1.5
Anisotropy Error 20 %
X (km)
Inaccurate Thomsen’s Parameters (VTI)
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5 10
Anisotropy Error 40 %
X (km)
3
4
Dep
th (
km
)
5 10
Anisotropy Error 10 %
X (km)5 10
Anisotropy Error 20 %
X (km)
Inaccurate Thomsen’s Parameters
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Contents Contents
Motivation
Anisotropy Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE VTI model
3-D SEG/EAGE VTI model
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0
4
Dep
th (
km
)0 5X (km) 0 5X (km)
VTI Aniso (y=1.5 km)Iso (y=1.5 km)
1.0045.0 00 00
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0
4
Dep
th (
km
)0 5Y (km) 0 5Y (km)
VTI Aniso (x=1.5 km)Iso (x=1.5 km)
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0
4
Dep
th (
km
)0 5Y (km) 0 5Y (km)
VTI Aniso (x=3 km)Iso (x=3 km)
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00
5
Y (
km
)
5X (km) 0 5X (km)
VTI Aniso (z=0.5 km)Iso (z=0.5 km)
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00
5
Y (
km
)
5X (km) 0 5X (km)
VTI Aniso (z=2.5 km)Iso (z=2.5 km)
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Contents Contents
Motivation
Anisotropy Wave-Equation Migration
Numerical Examples:
Cusp model
Conclusions
2-D SEG/EAGE model
3-D SEG/EAGE model
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Conclusions Conclusions
Works for 2-D and 3-D media
New > 78 Anisotropic wave propagator:
Improves spatial resolution
Valid for VTI and TI
o
78 Propagator Cost = Cost of Standard 45^o propagator
o
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Thanks To Thanks To
2003 UTAM Sponsors
CHPC